" The problem is that the Primitive region you are picking from is glued to another region and an error in the code is preventing the new region being correctly generated. This has been filed as a QA Defect and development will try and fix it in 13. The workaround is to open the Mesh > Connectivity branch in the tree - find the connection (this can be done using the search facilities (Control F) and typing in the name), deleting the connection, doing the primitive editing and then reconnecting regions as required or setting up a domain interface if preferred. Obviously it is important to remember the connectivity before attempting to do this."

So far I have done this:
I just created a standalone CFX module, and connected the Mesh from old one to Setup of the new one.
Since the new module does not know anything about previous geometry and surfaces, it generated them just fine.
Not sure if this is a good way to do this, but it works for now.

I also tried this, although it's probably not recommended:
I went to View->Files, found the file that corresponds to the setup cell which gives me problems, and deleted it.
This way, CFX pre had to start fresh, and thus problem has been removed.